To Create a Stone with Six Birds: Emergent Geometric and Thermodynamic Regimes from a Minimal Stochastic Substrate

Abstract

How much macroscopic regularity can a random Markov chain acquire when it is allowed to rewrite itself using only a small set of closure operations? We study this question in a minimal stochastic substrate: a random transition kernel on n states that is repeatedly rewritten, gated and reorganized by six primitive operations. Which of the 36 possible interactions among these primitives are switched on is recorded in a 6×6 enable matrix, the Primitive Interaction Closure Algebra (PICA), which turns a closure mechanism into a finite object that can be ablated cell by cell. After each run we coarse-grain the final kernel at a ladder of resolutions and audit the resulting macro kernels for structure, time asymmetry, least-action dominance, and geometric and diffusive regularity. Across 2,070 runs of a prescribed suite of 69 PICA configurations at n = 32, 64 and 128, plus a selected panel of 237 runs at n = 256, switching on the full set of interactions lowers a capped time-asymmetry measure far below that of a two-cell baseline at every size, while its effect on structure depends on size. Geometric regularity is protocol-dependent: an alternative coarse-graining lens shows consistent gains, but the default resolution ladder does not. Under one fixed coarse-graining instrument, a six-cell generator is sufficient, on a fresh seed panel, to produce low time asymmetry together with retained structure, and deleting one of its cells destroys this profile. At a coarse anchor resolution, probe rank correlations are strong, and a partition-competition effect on diffusion misfit changes sign with system size. These are finite-size, protocol-specific findings about a computational model, not physical laws; the paper states the scope of each. Version 2 (5 October 2026). This version follows a mathematical review and correction of the implementation. All reported data were regenerated with the corrected code, the Methods were brought into line with it, and the six claims were restated at the scope the regenerated evidence supports. A revision history is given in the paper's final appendix. Code and data: https://github.com/ioannist/six-birds-pica Keywords: Emergence; Coarse-graining; Markov chains; Stochastic thermodynamics; Irreversibility; Spectral methods; Diffusion maps; Six Birds Theory; Emergence Calculus